Revealing mistake: During the court martial scene, a directorial error led to Kirk's exit from the briefing room in the wrong direction. He walks off screen into what would have been a solid wall or corner. The briefing room's only door is on the other side of the room. (00:39:15)Jean G
Continuity mistake: After Garth attempts to convince Mr Scott that he is Captain Kirk and to beam him up, while he is banging his fists on the floor in anger he transforms back into himself and the fly like ring on his finger breaks and slides out of shot. It's quite hard to see but it does. The next time we see Garth's hand his ring is intact yet he has not transformed himself again to repair it.
Other mistake: Kirk's voice-over log entry says he's been "placed on the surface of an asteroid." The Metrons earlier referred to it as a planet prepared for them with a suitable atmosphere. How and why does Kirk suddenly assume that it's an asteroid rather than a full sized planet? As an experienced space veteran, Kirk, who has already been told that it's a planet, would never make such a mistake. (00:23:45)Jean G
Factual error: The Enterprise accidentally travels a short distance outside the galaxy and can't find its way back. But they'd have to travel for months to get so far outside the Milky Way that they couldn't, well, just turn around in the void and see it. Our galaxy is huge. 100,000 light years across. Very huge. And that barrier may surround the galaxy, but even it is big, pink and visible.Jean G